Population systems with nonlinear discrete diffusion terms.
The authors investigate the asymptotic behavior of the period population system with nonlinear discrete diffusion terms \[ \begin{aligned} x_i'&= D_i(t,x_1,\ldots,x_n)+ x_if_i(t,x_i,y_i),\\ y_i'&= E_i(t,y_1,\ldots,y_n)+y_ig_i(t,x_i,y_i), \end{aligned}\quad 1\leq i\leq n, \] where \(f_i,g_i: \mathbb{R}\times \mathbb{R}_+^2\rightarrow \mathbb{R}\) are continuous functions which are \(T\)-periodic in \(t\) and locally Lipschitz continuous in \((x_i,y_i)\) and \(D=(D_1,\ldots,D_n),\) \(E=(E_1,\ldots,E_n)\) are \(T\)-periodic nonlinear diffusion functions. For a predator-prey system, the authors prove by using Brower's degree theory, that there exists a positive \(T\)-periodic solution under some conditions and generalize the extinction result of \textit{J. López-Gómez, R. Ortega} and the second author [Adv. Differ. Equ. 1, 403--423 (1996; Zbl 0849.34026)]. For a competitive system, they present by using the method of monotone iterative schemes conditions for the global asymptotic stability of trivial and semitrivial \(T\)-periodic solutions, further, conditions for the existence of an unstable coexistence state are also investigated.
scientific article; zbMATH DE number 1956894